Disproved
Integral Hodge variant · counterexamples ↗
Deligne · official Hodge description · Section 2(iv), PDF page 2 →
Observed Oct 10, 2026
Conjecture · version 2
Integral Hodge variant · counterexamples
Earlier version 2. View current version 3 →
Statement
The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.
Assumptions
- Integral, rather than rational, coefficients.
- Smooth projective complex varieties; the assertion is existential.
Scope
Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.
Evidence
Source reports
Connections
In this problem · publication v9 · Selected context
Hodge conjecture →
Different statement · related variant · Disproved
Changing the coefficient ring changes the statement. Rational Hodge stays open; this refutation must never label the canonical Problem refuted.
Related statements
Integral coefficients ask for a stronger assertion and admit counterexamples, whereas the canonical question permits rational combinations.
An integral obstruction can disappear after tensoring with Q and does not refute rational Hodge.
Details & sources
rational Hodge holds in all codimensions →
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
- Smooth, projective and defined over C.
- Rational coefficients; the intersection is taken after complexification.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Integral Hodge variant · counterexamples →
Disproved · version 2
The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.
- Integral, rather than rational, coefficients.
- Smooth projective complex varieties; the assertion is existential.
Scope: Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.
Relationship not independently reviewed.
Connection in Hodge conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Deligne · official Hodge description →
- Section 2(iv), PDF page 2 → · Original passage ↗
Recorded wording
The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.
paraphrase · No upstream full text or code is republished by this record.
- §1, PDF page 1: rational cycle-class formulation → · Original passage ↗
Recorded wording
For every smooth projective complex variety X and every codimension p, CH^p(X)⊗Q maps onto H^{2p}(X,Q) ∩ H^{p,p}(X).
paraphrase · No upstream full text or code is republished by this record.
Pierre Deligne · Published
Official Clay problem description; historical exposition, not a current literature survey.
Official description PDF ↗Manuscript / exposition · external
Revision & attribution
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Official description PDF: official-description. External evidence linked; no OpenMath execution or independent reproduction is asserted.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Disproved
Integral Hodge variant · counterexamples ↗
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-integral-hodge-counterexamples
- Statement version
- 2
- SHA-256
- a79d61833ae0ea9a585f3df0059075223556d6206383f7ba0d880bea123e800c